Sarajevo is a city that gets asked about its worst years and would rather you came for the coffee. It sits in a narrow valley on the Miljacka river, shaped by two empires that left their marks a few steps apart: an Ottoman bazaar founded in 1462 and an Austro-Hungarian center built after the occupation of 1878. The tourist board's nickname for the old center is Little Jerusalem, because a mosque, an Orthodox church, a synagogue and a Catholic cathedral stand within a few hundred meters of each other. Then came the twentieth century: the shots of 28 June 1914 that started the First World War, the 1984 Winter Olympics, and the siege of 1992-96, the longest siege of a capital city in modern warfare.
The rest is more relaxed than that sounds. A coffee takes an hour, the ćevapi are small and very good, and the Olympic mountains, Jahorina and Bjelašnica, are about 30 km from the old town. Five days gives you three in the city and one on each mountain. The two mountain days need snow; the three city days work in any month, and the ski days can be swapped for the trips in the "more time" section.
Day one is the Ottoman bazaar, day two the Austro-Hungarian center and the story of 1914, day three the Trebević cable car, the bobsled track and the Tunnel of Hope, then Jahorina on day four and Bjelašnica on day five. Start on a Friday if you can: the cable car certainly runs from Thursday to Sunday, the museums that close on Sundays are done by then, and you ski on a Monday and Tuesday, when the slopes are quiet. For the country behind the city, see Bosnia and Herzegovina.